Can my insurance company be liable for neglect regarding a mold issue?

Question Details: About 3 months ago, we had a plumbing issue at our home. The outside water spigot cracked and had been leaking into the house. It soaked the carpet in our daughter's room and into the hallway. I did the repair myself and contacted our insurance company. We tried to use a shop vacuum to clean the water damaged rugs but it was no use. We pulled the carpets up and removed some furniture. The insurance company sent an adjuster to our place to inspect the damage. Using a moisture reader, the adjuster said that he detected moisture. The adjuster said that he was going to write up a report and our insurance company would get back to us. We did not hear anything from them for a couple of months. On the 8th of this month, I tried sending an email to them regarding the claim with no response. On the 22nd, I called the insurance company's claims department to see what was going on and was told that they forgot about us as they have been very busy. The insurance company said the damages that the adjuster saw total $5,000 and they were going to cut a check, minus the deductible, in order for us to make repairs. While talking to the gentleman from the claims department, he realized that no one was sent from a mold mitigation company to check for mold since it was a plumbing leak. So a company was scheduled and came out the same day. We do have mold growing on the walls of the bathroom and probably in the walls as well. We have a $10,000 mold/fungi clause in our policy. Since the insurance company neglected to send a repair or mold mitigation team sooner, are they now liable? For weeks the family has been complaining of headaches and such. I, myself have been very fatigued and with joint pain, possibly a result of mold in the home.

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